A large-deformation strain-gradient theory for isotropic viscoplastic materials

This study develops a thermodynamically consistent large-deformation theory of strain-gradient viscoplasticity for isotropic materials based on: (i) a scalar and a vector microstress consistent with a microforce balance; (ii) a mechanical version of the two laws of thermodynamics for isothermal conditions, that includes via the microstresses the work performed during viscoplastic flow; and (iii) a constitutive theory that allows:
• the free energy to depend on ∇γp∇γp, the gradient of equivalent plastic strain  γpγp, and this leads to the vector microstress having an energetic component;
• strain-hardening dependent on the equivalent plastic strain γpγp, and a scalar measure ϕpϕp related to the accumulation of geometrically necessary dislocations; and
• a dissipative part of the vector microstress to depend on ∇νp∇νp, the gradient of the equivalent plastic strain rate.The microscopic force balance, when augmented by constitutive relations for the microscopic stresses, results in a nonlocal flow rule in the form of a second-order partial differential equation for the equivalent plastic strain γpγp. In general, the flow rule, being nonlocal, requires microscopic boundary conditions. However, for problems which do not involve boundary conditions on γpγp, and for situations in which the dissipative part of the microstress may be neglected, the nonlocal flow rule may be inverted to give an equation for the plastic strain rate in the conventional form, but with additional gradient-dependent strengthening terms. For such special circumstances the theory may be relatively easily implemented by writing a user-material subroutine for standard finite element programs. We have implemented such a two-dimensional finite deformation plane–strain theory, and using this numerical capability we here report on our studies concerning: (a) the gradient-stabilization of shear band widths in problems which exhibit shear localization; (b) strengthening in pure bending due to strain-gradient effects; and (c) the well-known size-effect regarding hardness versus indentation depth in nano/micro-indentation experiments.